Metamath Proof Explorer


Theorem psgnfzto1stlem

Description: Lemma for psgnfzto1st . Our permutation of rank ( n + 1 ) can be written as a permutation of rank n composed with a transposition. (Contributed by Thierry Arnoux, 21-Aug-2020)

Ref Expression
Hypothesis psgnfzto1st.d D = 1 N
Assertion psgnfzto1stlem K K + 1 D i D if i = 1 K + 1 if i K + 1 i 1 i = pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i

Proof

Step Hyp Ref Expression
1 psgnfzto1st.d D = 1 N
2 ovex K + 1 V
3 ovex i 1 V
4 vex i V
5 3 4 ifex if i K + 1 i 1 i V
6 2 5 ifex if i = 1 K + 1 if i K + 1 i 1 i V
7 eqid i D if i = 1 K + 1 if i K + 1 i 1 i = i D if i = 1 K + 1 if i K + 1 i 1 i
8 6 7 fnmpti i D if i = 1 K + 1 if i K + 1 i 1 i Fn D
9 8 a1i K K + 1 D i D if i = 1 K + 1 if i K + 1 i 1 i Fn D
10 eqid pmTrsp D = pmTrsp D
11 1 10 pmtrto1cl K K + 1 D pmTrsp D K K + 1 ran pmTrsp D
12 eqid ran pmTrsp D = ran pmTrsp D
13 10 12 pmtrff1o pmTrsp D K K + 1 ran pmTrsp D pmTrsp D K K + 1 : D 1-1 onto D
14 f1ofn pmTrsp D K K + 1 : D 1-1 onto D pmTrsp D K K + 1 Fn D
15 11 13 14 3syl K K + 1 D pmTrsp D K K + 1 Fn D
16 simpr K K + 1 D i D i = 1 i = 1
17 16 iftrued K K + 1 D i D i = 1 if i = 1 K if i K i 1 i = K
18 simpl K K + 1 D K
19 18 nnred K K + 1 D K
20 fz1ssnn 1 N
21 1 eleq2i K + 1 D K + 1 1 N
22 21 biimpi K + 1 D K + 1 1 N
23 22 adantl K K + 1 D K + 1 1 N
24 20 23 sselid K K + 1 D K + 1
25 24 nnred K K + 1 D K + 1
26 elfz1b K + 1 1 N K + 1 N K + 1 N
27 26 simp2bi K + 1 1 N N
28 22 27 syl K + 1 D N
29 28 adantl K K + 1 D N
30 29 nnred K K + 1 D N
31 19 lep1d K K + 1 D K K + 1
32 elfzle2 K + 1 1 N K + 1 N
33 23 32 syl K K + 1 D K + 1 N
34 19 25 30 31 33 letrd K K + 1 D K N
35 29 nnzd K K + 1 D N
36 fznn N K 1 N K K N
37 35 36 syl K K + 1 D K 1 N K K N
38 18 34 37 mpbir2and K K + 1 D K 1 N
39 38 1 eleqtrrdi K K + 1 D K D
40 39 ad2antrr K K + 1 D i D i = 1 K D
41 17 40 eqeltrd K K + 1 D i D i = 1 if i = 1 K if i K i 1 i D
42 simpr K K + 1 D i D ¬ i = 1 ¬ i = 1
43 42 iffalsed K K + 1 D i D ¬ i = 1 if i = 1 K if i K i 1 i = if i K i 1 i
44 simpr K K + 1 D i D ¬ i = 1 i K i K
45 44 iftrued K K + 1 D i D ¬ i = 1 i K if i K i 1 i = i 1
46 42 adantr K K + 1 D i D ¬ i = 1 i K ¬ i = 1
47 1 20 eqsstri D
48 simpllr K K + 1 D i D ¬ i = 1 i K i D
49 47 48 sselid K K + 1 D i D ¬ i = 1 i K i
50 nn1m1nn i i = 1 i 1
51 49 50 syl K K + 1 D i D ¬ i = 1 i K i = 1 i 1
52 51 ord K K + 1 D i D ¬ i = 1 i K ¬ i = 1 i 1
53 46 52 mpd K K + 1 D i D ¬ i = 1 i K i 1
54 53 nnred K K + 1 D i D ¬ i = 1 i K i 1
55 49 nnred K K + 1 D i D ¬ i = 1 i K i
56 30 ad3antrrr K K + 1 D i D ¬ i = 1 i K N
57 55 lem1d K K + 1 D i D ¬ i = 1 i K i 1 i
58 48 1 eleqtrdi K K + 1 D i D ¬ i = 1 i K i 1 N
59 elfzle2 i 1 N i N
60 58 59 syl K K + 1 D i D ¬ i = 1 i K i N
61 54 55 56 57 60 letrd K K + 1 D i D ¬ i = 1 i K i 1 N
62 53 61 jca K K + 1 D i D ¬ i = 1 i K i 1 i 1 N
63 fznn N i 1 1 N i 1 i 1 N
64 35 63 syl K K + 1 D i 1 1 N i 1 i 1 N
65 64 ad3antrrr K K + 1 D i D ¬ i = 1 i K i 1 1 N i 1 i 1 N
66 62 65 mpbird K K + 1 D i D ¬ i = 1 i K i 1 1 N
67 66 1 eleqtrrdi K K + 1 D i D ¬ i = 1 i K i 1 D
68 45 67 eqeltrd K K + 1 D i D ¬ i = 1 i K if i K i 1 i D
69 simpr K K + 1 D i D ¬ i = 1 ¬ i K ¬ i K
70 69 iffalsed K K + 1 D i D ¬ i = 1 ¬ i K if i K i 1 i = i
71 simpllr K K + 1 D i D ¬ i = 1 ¬ i K i D
72 70 71 eqeltrd K K + 1 D i D ¬ i = 1 ¬ i K if i K i 1 i D
73 68 72 pm2.61dan K K + 1 D i D ¬ i = 1 if i K i 1 i D
74 43 73 eqeltrd K K + 1 D i D ¬ i = 1 if i = 1 K if i K i 1 i D
75 41 74 pm2.61dan K K + 1 D i D if i = 1 K if i K i 1 i D
76 75 ralrimiva K K + 1 D i D if i = 1 K if i K i 1 i D
77 eqid i D if i = 1 K if i K i 1 i = i D if i = 1 K if i K i 1 i
78 77 fnmpt i D if i = 1 K if i K i 1 i D i D if i = 1 K if i K i 1 i Fn D
79 76 78 syl K K + 1 D i D if i = 1 K if i K i 1 i Fn D
80 77 rnmptss i D if i = 1 K if i K i 1 i D ran i D if i = 1 K if i K i 1 i D
81 76 80 syl K K + 1 D ran i D if i = 1 K if i K i 1 i D
82 fnco pmTrsp D K K + 1 Fn D i D if i = 1 K if i K i 1 i Fn D ran i D if i = 1 K if i K i 1 i D pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i Fn D
83 15 79 81 82 syl3anc K K + 1 D pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i Fn D
84 simpr K K + 1 D x D x = 1 x = 1
85 84 iftrued K K + 1 D x D x = 1 if x = 1 K if x K x 1 x = K
86 85 fveq2d K K + 1 D x D x = 1 pmTrsp D K K + 1 if x = 1 K if x K x 1 x = pmTrsp D K K + 1 K
87 fzfi 1 N Fin
88 1 87 eqeltri D Fin
89 88 a1i K K + 1 D D Fin
90 23 21 sylibr K K + 1 D K + 1 D
91 19 ltp1d K K + 1 D K < K + 1
92 19 91 ltned K K + 1 D K K + 1
93 10 pmtrprfv D Fin K D K + 1 D K K + 1 pmTrsp D K K + 1 K = K + 1
94 89 39 90 92 93 syl13anc K K + 1 D pmTrsp D K K + 1 K = K + 1
95 94 ad2antrr K K + 1 D x D x = 1 pmTrsp D K K + 1 K = K + 1
96 86 95 eqtr2d K K + 1 D x D x = 1 K + 1 = pmTrsp D K K + 1 if x = 1 K if x K x 1 x
97 88 a1i K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 D Fin
98 39 ad4antr K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 K D
99 90 ad4antr K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 K + 1 D
100 92 ad4antr K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 K K + 1
101 10 pmtrprfv2 D Fin K D K + 1 D K K + 1 pmTrsp D K K + 1 K + 1 = K
102 97 98 99 100 101 syl13anc K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 pmTrsp D K K + 1 K + 1 = K
103 91 ad4antr K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 K < K + 1
104 simpr K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 x = K + 1
105 103 104 breqtrrd K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 K < x
106 19 ad4antr K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 K
107 simpr K K + 1 D x D x D
108 47 107 sselid K K + 1 D x D x
109 108 nnred K K + 1 D x D x
110 109 ad3antrrr K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 x
111 106 110 ltnled K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 K < x ¬ x K
112 105 111 mpbid K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 ¬ x K
113 112 iffalsed K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 if x K x 1 x = x
114 113 104 eqtrd K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 if x K x 1 x = K + 1
115 114 fveq2d K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 pmTrsp D K K + 1 if x K x 1 x = pmTrsp D K K + 1 K + 1
116 104 oveq1d K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 x 1 = K + 1 - 1
117 106 recnd K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 K
118 1cnd K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 1
119 117 118 pncand K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 K + 1 - 1 = K
120 116 119 eqtrd K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 x 1 = K
121 102 115 120 3eqtr4rd K K + 1 D x D ¬ x = 1 x K + 1 x = K + 1 x 1 = pmTrsp D K K + 1 if x K x 1 x
122 simplr K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x K + 1
123 simpr K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x K + 1
124 123 necomd K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K + 1 x
125 109 ad3antrrr K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x
126 25 ad4antr K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K + 1
127 125 126 ltlend K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x < K + 1 x K + 1 K + 1 x
128 122 124 127 mpbir2and K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x < K + 1
129 108 ad3antrrr K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x
130 simpll K K + 1 D x D K
131 130 ad3antrrr K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K
132 nnleltp1 x K x K x < K + 1
133 129 131 132 syl2anc K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x K x < K + 1
134 128 133 mpbird K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x K
135 134 iftrued K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 if x K x 1 x = x 1
136 135 fveq2d K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 pmTrsp D K K + 1 if x K x 1 x = pmTrsp D K K + 1 x 1
137 88 a1i K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 D Fin
138 39 ad4antr K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K D
139 simp-5r K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K + 1 D
140 simpr K K + 1 D x D ¬ x = 1 ¬ x = 1
141 140 ad2antrr K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 ¬ x = 1
142 elnn1uz2 x x = 1 x 2
143 129 142 sylib K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x = 1 x 2
144 143 ord K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 ¬ x = 1 x 2
145 141 144 mpd K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 2
146 uz2m1nn x 2 x 1
147 145 146 syl K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1
148 139 28 syl K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 N
149 147 nnred K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1
150 131 139 30 syl2anc K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 N
151 125 lem1d K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1 x
152 107 ad3antrrr K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x D
153 152 1 eleqtrdi K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1 N
154 elfzle2 x 1 N x N
155 153 154 syl K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x N
156 149 125 150 151 155 letrd K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1 N
157 147 148 156 3jca K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1 N x 1 N
158 elfz1b x 1 1 N x 1 N x 1 N
159 157 158 sylibr K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1 1 N
160 159 1 eleqtrrdi K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1 D
161 138 139 160 3jca K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K D K + 1 D x 1 D
162 131 139 92 syl2anc K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K K + 1
163 simpr K K + 1 D x D ¬ x = 1 x K + 1 K = x 1 K = x 1
164 163 oveq1d K K + 1 D x D ¬ x = 1 x K + 1 K = x 1 K + 1 = x - 1 + 1
165 109 recnd K K + 1 D x D x
166 165 ad3antrrr K K + 1 D x D ¬ x = 1 x K + 1 K = x 1 x
167 1cnd K K + 1 D x D ¬ x = 1 x K + 1 K = x 1 1
168 166 167 npcand K K + 1 D x D ¬ x = 1 x K + 1 K = x 1 x - 1 + 1 = x
169 164 168 eqtr2d K K + 1 D x D ¬ x = 1 x K + 1 K = x 1 x = K + 1
170 169 ex K K + 1 D x D ¬ x = 1 x K + 1 K = x 1 x = K + 1
171 170 necon3d K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K x 1
172 171 imp K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K x 1
173 149 125 126 151 128 lelttrd K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1 < K + 1
174 149 173 ltned K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1 K + 1
175 174 necomd K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K + 1 x 1
176 162 172 175 3jca K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 K K + 1 K x 1 K + 1 x 1
177 10 pmtrprfv3 D Fin K D K + 1 D x 1 D K K + 1 K x 1 K + 1 x 1 pmTrsp D K K + 1 x 1 = x 1
178 137 161 176 177 syl3anc K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 pmTrsp D K K + 1 x 1 = x 1
179 136 178 eqtr2d K K + 1 D x D ¬ x = 1 x K + 1 x K + 1 x 1 = pmTrsp D K K + 1 if x K x 1 x
180 121 179 pm2.61dane K K + 1 D x D ¬ x = 1 x K + 1 x 1 = pmTrsp D K K + 1 if x K x 1 x
181 109 ad2antrr K K + 1 D x D ¬ x = 1 x K x
182 19 ad3antrrr K K + 1 D x D ¬ x = 1 x K K
183 25 ad3antrrr K K + 1 D x D ¬ x = 1 x K K + 1
184 simpr K K + 1 D x D ¬ x = 1 x K x K
185 31 ad3antrrr K K + 1 D x D ¬ x = 1 x K K K + 1
186 181 182 183 184 185 letrd K K + 1 D x D ¬ x = 1 x K x K + 1
187 186 ex K K + 1 D x D ¬ x = 1 x K x K + 1
188 187 con3d K K + 1 D x D ¬ x = 1 ¬ x K + 1 ¬ x K
189 188 imp K K + 1 D x D ¬ x = 1 ¬ x K + 1 ¬ x K
190 189 iffalsed K K + 1 D x D ¬ x = 1 ¬ x K + 1 if x K x 1 x = x
191 190 fveq2d K K + 1 D x D ¬ x = 1 ¬ x K + 1 pmTrsp D K K + 1 if x K x 1 x = pmTrsp D K K + 1 x
192 88 a1i K K + 1 D x D ¬ x = 1 ¬ x K + 1 D Fin
193 39 ad3antrrr K K + 1 D x D ¬ x = 1 ¬ x K + 1 K D
194 90 ad3antrrr K K + 1 D x D ¬ x = 1 ¬ x K + 1 K + 1 D
195 107 ad2antrr K K + 1 D x D ¬ x = 1 ¬ x K + 1 x D
196 193 194 195 3jca K K + 1 D x D ¬ x = 1 ¬ x K + 1 K D K + 1 D x D
197 92 ad3antrrr K K + 1 D x D ¬ x = 1 ¬ x K + 1 K K + 1
198 19 ad3antrrr K K + 1 D x D ¬ x = 1 ¬ x K + 1 K
199 25 ad3antrrr K K + 1 D x D ¬ x = 1 ¬ x K + 1 K + 1
200 109 ad2antrr K K + 1 D x D ¬ x = 1 ¬ x K + 1 x
201 91 ad3antrrr K K + 1 D x D ¬ x = 1 ¬ x K + 1 K < K + 1
202 simpr K K + 1 D x D ¬ x = 1 ¬ x K + 1 ¬ x K + 1
203 199 200 ltnled K K + 1 D x D ¬ x = 1 ¬ x K + 1 K + 1 < x ¬ x K + 1
204 202 203 mpbird K K + 1 D x D ¬ x = 1 ¬ x K + 1 K + 1 < x
205 198 199 200 201 204 lttrd K K + 1 D x D ¬ x = 1 ¬ x K + 1 K < x
206 198 205 ltned K K + 1 D x D ¬ x = 1 ¬ x K + 1 K x
207 199 204 ltned K K + 1 D x D ¬ x = 1 ¬ x K + 1 K + 1 x
208 197 206 207 3jca K K + 1 D x D ¬ x = 1 ¬ x K + 1 K K + 1 K x K + 1 x
209 10 pmtrprfv3 D Fin K D K + 1 D x D K K + 1 K x K + 1 x pmTrsp D K K + 1 x = x
210 192 196 208 209 syl3anc K K + 1 D x D ¬ x = 1 ¬ x K + 1 pmTrsp D K K + 1 x = x
211 191 210 eqtr2d K K + 1 D x D ¬ x = 1 ¬ x K + 1 x = pmTrsp D K K + 1 if x K x 1 x
212 180 211 ifeqda K K + 1 D x D ¬ x = 1 if x K + 1 x 1 x = pmTrsp D K K + 1 if x K x 1 x
213 140 iffalsed K K + 1 D x D ¬ x = 1 if x = 1 K if x K x 1 x = if x K x 1 x
214 213 fveq2d K K + 1 D x D ¬ x = 1 pmTrsp D K K + 1 if x = 1 K if x K x 1 x = pmTrsp D K K + 1 if x K x 1 x
215 212 214 eqtr4d K K + 1 D x D ¬ x = 1 if x K + 1 x 1 x = pmTrsp D K K + 1 if x = 1 K if x K x 1 x
216 96 215 ifeqda K K + 1 D x D if x = 1 K + 1 if x K + 1 x 1 x = pmTrsp D K K + 1 if x = 1 K if x K x 1 x
217 eqidd K K + 1 D x D i D if i = 1 K if i K i 1 i = i D if i = 1 K if i K i 1 i
218 eqeq1 i = x i = 1 x = 1
219 breq1 i = x i K x K
220 oveq1 i = x i 1 = x 1
221 id i = x i = x
222 219 220 221 ifbieq12d i = x if i K i 1 i = if x K x 1 x
223 218 222 ifbieq2d i = x if i = 1 K if i K i 1 i = if x = 1 K if x K x 1 x
224 223 adantl K K + 1 D x D i = x if i = 1 K if i K i 1 i = if x = 1 K if x K x 1 x
225 ovex x 1 V
226 vex x V
227 225 226 ifcli if x K x 1 x V
228 227 a1i K K + 1 D x D if x K x 1 x V
229 130 228 ifexd K K + 1 D x D if x = 1 K if x K x 1 x V
230 217 224 107 229 fvmptd K K + 1 D x D i D if i = 1 K if i K i 1 i x = if x = 1 K if x K x 1 x
231 230 fveq2d K K + 1 D x D pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i x = pmTrsp D K K + 1 if x = 1 K if x K x 1 x
232 216 231 eqtr4d K K + 1 D x D if x = 1 K + 1 if x K + 1 x 1 x = pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i x
233 breq1 i = x i K + 1 x K + 1
234 233 220 221 ifbieq12d i = x if i K + 1 i 1 i = if x K + 1 x 1 x
235 218 234 ifbieq2d i = x if i = 1 K + 1 if i K + 1 i 1 i = if x = 1 K + 1 if x K + 1 x 1 x
236 225 226 ifex if x K + 1 x 1 x V
237 2 236 ifex if x = 1 K + 1 if x K + 1 x 1 x V
238 235 7 237 fvmpt x D i D if i = 1 K + 1 if i K + 1 i 1 i x = if x = 1 K + 1 if x K + 1 x 1 x
239 238 adantl K K + 1 D x D i D if i = 1 K + 1 if i K + 1 i 1 i x = if x = 1 K + 1 if x K + 1 x 1 x
240 funmpt Fun i D if i = 1 K if i K i 1 i
241 240 a1i K K + 1 D x D Fun i D if i = 1 K if i K i 1 i
242 76 adantr K K + 1 D x D i D if i = 1 K if i K i 1 i D
243 dmmptg i D if i = 1 K if i K i 1 i D dom i D if i = 1 K if i K i 1 i = D
244 242 243 syl K K + 1 D x D dom i D if i = 1 K if i K i 1 i = D
245 107 244 eleqtrrd K K + 1 D x D x dom i D if i = 1 K if i K i 1 i
246 fvco Fun i D if i = 1 K if i K i 1 i x dom i D if i = 1 K if i K i 1 i pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i x = pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i x
247 241 245 246 syl2anc K K + 1 D x D pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i x = pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i x
248 232 239 247 3eqtr4d K K + 1 D x D i D if i = 1 K + 1 if i K + 1 i 1 i x = pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i x
249 9 83 248 eqfnfvd K K + 1 D i D if i = 1 K + 1 if i K + 1 i 1 i = pmTrsp D K K + 1 i D if i = 1 K if i K i 1 i